Neither side had been able to find a breakthrough in normal time, and after five penalties each there was still nothing to separate the sides with both successfully converting three apiece.
Teenager Stuart McKinstry scored for Leeds to make it 6-5, but Fulham’s Rodrigo Muniz was unable to convert, leaving Leeds to progress.
“It was difficult for us. We conceded too many chances to the opponent,” Bielsa said. “It was difficult for all our players.”
McKinstry and Joe Gelhardt, 19, were selected by Bielsa to take penalties in the shoot-out, with both successful.
“They showed personality and character,” he said. “Gelhardt had missed his last two penalties and took on board the responsibility and managed to score and McKinstry also had to take one in the most decisive moments and they showed character.”
Marco Silva believes his Fulham side deserved more from the match itself but could not fault his players.
“When I made 11 changes in the starting XI and I saw our team match our opponent like we did this evening - of course I’m not happy with the result - like all of us I think we deserved something more from the game,” Silva said.
“But the result is what it is, but we have to be proud of our players and what they did and how they played for the 95 minutes, with very good intensity in some moments of the game.
“Unfortunately in the penalties they were more competent than us, because for me it’s not just a matter of luck or not.
“But I’m proud of what we did and our players.
“The ball went to the crossbar of the post and we lost in penalties as well, but by working the way that we are doing and the way that we will keep doing, the good moments will come for us.”